Today’s Globe and Mail featured a column by Gary Mason on a world without oil. “If you believe that the economy is structured in such a way that it needs to grow continually in order to survive,” it states, “then it will take an endless supply of energy to feed it. ” The article then raises the question, “How does an economy grow exponentially forever if the one element it needs more than anything to flourish is contracting with time?” This is a common refrain from environmentalists such as David Suzuki (here, here, here and likely a thousand other places): “it’s absurd to rely on economies based on constant growth on a finite planet.” But, is it? I’ll have more on this at Macleans in a couple of days, but this will serve as a technical primer.

Intuitively, it sounds simple – if I use up a certain amount of a finite quantity each year, it will eventually run out. But that tells you that you can’t have constant or increasing resource extraction from a finite resource, it doesn’t tell you anything about what you do with the resources you extract, how productive they are, or whether or not they enable continued economic growth. It’s certainly possible to sustain exponential growth infinitely with finite resources, as long as productivity improves.

Let me take you through an example (this is a really basic model, but I’ve fit it with some reasonable numbers so its intuitive). Suppose that gross world product (real, including all environmental costs) is given by 1450*R*X, where R is resource productivity and X is extraction. If you use oil extraction as a proxy for resources, and we extract about 31.4 billion barrels of oil per year, and let R equal 1, you’ll get a gross world product of $45,515 billion, about the same as the CIA World Factbook estimate of 2012 gross world product. Let’s also suppose, for the sake of this argument, that the 1.8 trillion barrels of oil in current global reserves represents the sum total of all the oil which will ever be extracted – a finite resource.

With those numbers, the myopic approach to maintaining constant growth with no change in productivity would lead to all oil resources being exhausted in 55 years, and then instant economic collapse.

Myopic resource extraction

Of course, this would not actually happen, since prices would adjust even if there were no productivity changes. To understand what would happen, go to the last period before the collapse – a period in which the world extracts 35 billion barrels of oil out of a remaining stock of about 40 billion barrels. Knowing what was going to happen if you stuck with that plan, you’d likely decide that it makes sense to carry some extra oil through to the following year, to stave off collapse and/or to profit from absurdly high prices. In doing so, you’d raise prices in that year. Of course, people would have seen this coming too, leading to conservation of oil from previous years as well. This is a clumsy explanation of what Harold Hotelling wrote down almost 100 years ago – that since oil is like a capital asset, owners will act to maximize returns and this will smooth price and extraction decisions over time. If you imposed a Hotelling solution – one which maximized the value of oil over time, you’d end up with something which looks something like this:

Smoothed resource extraction

However, Hotelling doesn’t get you to economic growth with finite resources – production is still decreasing over time, and tends asymptotically to zero – it’s just that there is no collapse and oil is distributed over time such that there are no gains in net present value to be achieved by shifting production forward or back in time. (In the graph above, I approximated a 400 year solution – I didn’t solve the full optimal control problem).

If you want to get to increasing economic growth with a finite resource, you need an increase in productivity. Suppose that you still have the same finite resource stock, but that you become 3% more productive each year in your use of resources – you generate 3% more total product from each unit of resource extraction. The growth in productivity allows you to use fewer resources each year, while still increasing production. Resource stocks still decline, and approach zero asymptotically, but it’s like going half the distance to the goal line in football – you’ll get closer every time but you’ll never score.

Resource extraction with increasing productivity

So, how do you increase productivity? Energy is used in our economy as a complement to labour and capital, so if you want to increase the productivity of your finite resource then increase energy efficiency, decrease the resource-intensity of energy, increase labour productivity, or increase the quality of your human and physical capital. This is what Queen’s University economist John Hartwick had in mind when he wrote down the Hartwick rule – the mathematical proof of what I’ve just tried to do in words: as long as you invest sufficiently in improvements in productivity, and manage resources optimally, its possible to sustain infinite growth from a finite resource. Of course, the Hartwick rule is not a law – it doesn’t guarantee that this will always be achieved, and it certainly doesn’t say that it can be accomplished with any level of investment – it just tells you that its mathematically possible.

Saying that it’s impossible to achieve exponential growth infinitely with finite resources does nothing to advance our discussions of resource management and ignores plenty of evidence to the contrary in the economics literature. What we should be discussing instead is how to make sure we follow Hartwick’s rule, but that’s another story for another day.

28 responses to “Finite Resources and Infinite Growth”

As a Chemist I would ask a question about your argument. As scarcity becomes an issue and petroleum prices rise we will realize that using petroleum products for their energy content is not an optimum use. The last drop of petroleum refined will not be used to power the future’s version of a Honda,it will be cracked and/or combined into critical petrochemicals with a portion being used to produce a then ultra-expensive drug/chemical and the remainder turned into inputs for a future 3D printer to create some ultra-valuable widget/valve (since it will be the last one created using the technology).

In the current resource scenario (with ample relatively cheap oil) the limited volume, high-value uses of oil do not really make a dent in the market, but as supply dwindles it will become a much more significant part of the equation (and input cost into the final products).

Does the presence of a competing, alternative, high-value use for the resource simply change the rate of the curve, move the curve to the left or do something altogether different at some critical price-point?

Likely, that will be the case – we will shift uses of petroleum to places where its attributes are more valuable as substitutes become viable for other uses at lower costs. Since the curve is not really quantitative in any real sense, I can’t really say what the impact would be. Generally, if you think of substitutes for fossil energy, they allow higher production per unit fossil fuels, which is basically what the Hartwick rule says you need to achieve.

Absurd. If my paycheck is cut in half every month, I can still increase my standard of living forever? I’m sure you can make the math work out, since we know from calculus 1A that Infinity * 0 = 7. The problem lies in the premise of eternal exponential growth of productivity. How is that possible, especially while the resource base declines exponentially? You can’t make more and more out of basically nothing. Anyone can demand an increase in productivity, but that doesn’t make it happen. This sounds like armchair philosophy to me. Yes, the GWP continues to increase, but so does the human ecological footprint. Since nature tends to be nonlinear, you could have short-term growth in productivity even with a decline in resources, but infinite growth is just a Utopian fantasy. The only kind of “infinite” growth I can imagine is if the resource supply is constant in time, i.e. renewable resources like solar energy, and the GWP approaches a maximum possible value asymptotically. Or, in the language of ecology, the human ecological footprint approaches Earth’s carrying capacity for humans asymptotically.

It’s not armchair philosophy, it’s math. I can’t speak to why your paycheck declines every month, but there are many reasons that could occur while aggregate real productivity and real output increases. What I wrote says nothing about distribution. And yes, you are correct that part of the picture will be other forms of energy, part will be increased productivity of energy, and part will be increased productivity of labour and energy due to accumulated capital.

I think you misunderstood my paycheck analogy. My paycheck is analogous to oil as a resource in your example. An economy that runs mostly on oil and has decreasing amounts available to it, is like a person who gets a smaller paycheck every month. Of course, especially if he was wasteful before, he can still improve his standard of living by being smarter about how he spends his money, but only for so long. Once he is down to one dollar, not only will his standard of living cease to increase, he will not even be able to survive.

I’m not an economist, so I had to read up on Hartwick’s rule, but nowhere did I see that it was used to justify the concept of infinite growth based on a dwindling resource stock. Here’s a pretty good summary: http://www.hartwick.ca. To me this rule makes sense, but you have to realize what its limitations are. One major limiting factor is the fact that produced capital is a complex structure that requires a minimal, nontrivial amount of resources to maintain. The more produced capital you have, the more resources you need. So the claim that an increasing amount of capital goods can be maintained with resource stocks approaching nothing violates the laws of thermodynamics. As a matter of fact, you wouldn’t even be able to maintain production at a constant level.

Here’s the parallel to your paycheck analogy – yes, if someone has only one job, refuses to change jobs, or retrain, and accepts that their paycheck may decline, they will end up as the first graph in the piece or, at best, the second one. If, on the other hand, the person is able to source other means of income, from a different job in the same field, a different field altogether, or is able to adjust his/her lifestyle in such a way that utility from consumption increases regardless of decline in salary. So, yes, if you assume that the world is and always will be only dependent on oil as an energy source, that there are no reliable substitutes, and efficiency and/or share of economic production tied to oil can’t change, you’re right. If you don’t, you’re not.

Let’s look at the example of whale oil, or wood, or coal. All of these resources, at one point or another, where critical to all or most economic production. Our consumption for industrial purposes of whale oil has declined, and our share of production associated with wood and coal has declined as well. None of these would have been contemplated, or able to be backed up empirically, in advance.

Keep in mind what Hartwick’s seminal paper talked about – how to maintain constant per capita consumption with a finite resource. If you over-invest relative to that rule in capital assets, you would see increasing per capita consumption, and vice versa. Hartwick’s AER paper is the limiting condition, and I am changing it a bit here. I’ll take some license there, as I used to work as a research assistant for someone who knows a little bit about the Hartwick rule.

You’re contradicting yourself. In the article you said oil is a proxy for resources. Now you’re saying that if we run out of one resource we can just switch to another and thus maintain economic growth. Of course that’s possible, but it’s also obvious. Everybody knows that. But that wasn’t the point of your article, and that’s not what Hartwick’s rule says. You said that we can increase production forever even as resource stocks decline and approach zero asymptotically. As I explained above, that violates the laws of nature. There’s no way around it, and I say this as a physicist and ecologist.

Re-read Hartwick. Hartwick talks about investments in capital which increase the productivity of a particular resource within a stylized global production function, so no, I am not contradicting myself. Also, you might consider that I am better placed than you to understand the point of my article. Perhaps that’s not how you understood it, for which I take some responsibility, but to suggest that you know what I was actually trying to say but that I am now saying something different is not really a productive discussion to have.

“Resource stocks still decline, and approach zero asymptotically, but it’s like going half the distance to the goal line in football – you’ll get closer every time but you’ll never score.”

This is a Zeno’s paradox redux. One tacit premise of this is that the units are divisible ad infinitum. In other words, this proof works in a math that uses real numbers, but its application to the world we live in is actually a reductio ad absurdum argument against the idea that you can “divide the distance in half” infinitely.

The divisibility is actually not as important as the growth on the numerator – think of economic output per unit whale oil consumed – whether or not you can subdivide whale oil to a sufficiently small quantity is not really the issue.

Taking a Twitter pause to re-evaluate. Not sure if I’ll be back or not.

I don’t know, in the same way as people would not have contemplated mass distribution of e-books via PDF files on the internet while carving on stone tablets, I don’t think we can say. Perhaps there are physical limits to the degree of consumption we can provide with our renewable and non-renewable energy endowments? If I were making the argument that you can produce an infinite amount of energy with a finite amount of energy, as many people seem to have taken my argument, that would be clearly false. The issue is how energy is used within our economy. So, can we get more efficient forever? Somehow I find continuous improvement more easy to visualize than us hitting a point where, in all dimensions, nothing can be improved upon in any way.

Well, that’s a bit of a false dichotomy. We don’t have to choose between infinite improvements in efficiency (which you require as a premise, half of the time) and “nothing can be improved upon.”

I just noticed that you also published an elaboration of this in Maclean’s. I think you’re underlying point that people should be wary of underestimating increased productivity is fair.

Put another way, you are advocating for both strong and weak versions of your thesis, and I can probably get on board the weak version. But everyone will, quite understandably, respond to the strong version.

I wish you wrote more about the costs and risks of carbon destabilization.

It’s not a false dichotomy in that either we will continue to improve or we won’t. It’s false to suggest that any continuous improvement in productivity would imply a continuous improvement in economic well-being – that’s the basic point of the Hartwick rule. Hartwick ignores uncertainty, and says that, if we knew the means by which investing in capital could improve productivity, here’s what you’d have to do to ensure constant per capita consumption. That’s it. No more, no less. The reason I think it’s important is that it moves you from the “easy” soundbites of, “you can’t have infinite growth with finite resources” to a conversation about the degree to which productivity improvements of the sort envisioned by Hartwick are possible. It’s funny – most of the attacks on this piece are from people who care a lot about the environment, and have missed the fact that, basically, what it says is what they say all the time – you have to reduce economic dependence on oil and other finite resources.

I write about what grabs me when I have time to write – that’s about it.

If your point was that we “have to reduce economic dependence on oil and other finite resources”, then you should have just said this. Of course you’re getting flack from a certain set of people. Funny that you can’t tell why.

You mean I should have written something like, ” if you want to increase the productivity of your finite resource then increase energy efficiency and/or decrease the resource-intensity of energy?” I’ll try to fit that in next time.

I saw this on Revkin’s FaceBook page and thought it (and the MacLeans article) an interesting and misleading oversimplification, so I wrote a correction (or debate) and emailed it to you. You didn’t respond. perhaps you haven’t seen it yet. Here is the link.

Thanks for taking the time to respond at such length. I’m actually surprised that people would see any way to take this as a justification for burning as much oil and coal as we can, when it’s essentially the exact opposite. As you say in your piece, “to get to where Leach says we can go, we’ll actually have to become more sustainable in all we do.” That’s basically the point – if you think of the limit condition of Hartwick, it’s an economy when you are able to maintain constant per capita consumption while using less and less resource, something which we could only accomplish by both finding new ways to generate energy, making more efficient use of the energy we do produce, and ensuring that we eliminate uses of energy which generate net costs. I would expect that we don’t disagree as much as your piece implies with respect to what that looks like.

The degree to which our economy “depends” on oil, is certainly debatable in the long term, but not in the short term. In the short term, oil shortages are indeed highly disruptive, whereas the long term would see an entirely different story. If you haven’t done so, read Jevons on the exhaustion of coal on the late 19th century. Not only was he spectacularly wrong about coal, the more important part of the argument was that in which he argues that oil represents a poor substitute for coal and that the industrial economy depends on it and it alone. I expect that similar arguments were made with respect to other productive by finite resources over time.

Thanks for taking the time to respond at such length. I’m actually surprised that people would see any way to take this as a justification for burning as much oil and coal as we can, when it’s essentially the exact opposite. As you say in your piece, “to get to where Leach says we can go, we’ll actually have to become more sustainable in all we do.” That’s basically the point – if you think of the limit condition of Hartwick, it’s an economy when you are able to maintain constant per capita consumption while using less and less resource, something which we could only accomplish by both finding new ways to generate energy, making more efficient use of the energy we do produce, and ensuring that we eliminate uses of energy which generate net costs. I would expect that we don’t disagree as much as your piece implies with respect to what that looks like.

The degree to which our economy “depends” on oil, is certainly debatable in the long term, but not in the short term. In the short term, oil shortages are indeed highly disruptive, whereas the long term would see an entirely different story. If you haven’t done so, read Jevons on the exhaustion of coal on the late 19th century. Not only was he spectacularly wrong about coal, the more important part of the argument was that in which he argues that oil represents a poor substitute for coal and that the industrial economy depends on it and it alone. I expect that similar arguments were made with respect to other productive by finite resources over time.

OK, like I said in my response, we can probably agree on the application of Hotelling. (Hartwick is an addition to Hotelling, so I will continue to refer to the original. It’s the asymptotic Hotelling trajectory that does the intellectual “work” you want done. Apologies to Canadian pride.) I would just put more emphasis on the details of the thinking, engineering, and finance (including subsidy) that goes into advancing beyond fossil energy.

I think you gloss over my other, possibly more important point, which is the effect to which your attenuated explanation has on the politics and public discourse. You published an article in Macleans blog which, if misread (and it almost certainly will be), essentially justifies oil sands exploitation on the grounds of Hotelling’s rule. Not a word about externalities.

Not a note about Alberta’s intellectually crippling conflict of interest. I think this is a fairly clear case of denial.

I’m going to copy over the comments to my own blog, so my students can follow this.

Sure, it all needs a lot more detail. The point which I was trying to make is that having an economy which uses significant amounts of a finite resource, and continues to use some declining amount of that resource over time, can continue to grow – that’s also Hartwick’s point. For what it’s worth, you can get the Hartwick result without the Hotelling result holding, as long as you have fungible capital – you could have constant per capita consumption with a finite resource even if the extraction of that resource were not strictly optimal from an intertemporal perspective. If you don’t have fungible capital, you can still make it work in the limit, but you’d have consumption Euler equations which don’t hold over time.

With respect to the oil sands, I have no idea how you can read the Maclean’s piece as justifying the extraction of the entire oil sands resource on the grounds of the Hotelling rule, or how that would imply ignoring externalities. At least in so far as the Hartwick implementation is concerned, it’s a model of per capita consumption, and so any costs which are “external” to production decisions would remain internal to consumption, so any reasonable implementation of the Hartwick rule would take them into account. If you take the time to read my writing on the subject, I think you’ll be hard-pressed to find support for a “dig it all up and screw everyone else” approach to oil sands.

Andrew, it’s not me that’s misreading the piece. I’m fully familiar with the theory, possibly more so than you since I was made to learn the ecological economics criticism too.

It’s the people making comments on the Twittersphere and the comments on Maclean’s blog that are misreading your piece. This should be obvious, and I find it funny that you can’t see this.

The reason, which is clear to me, if not to you, is that your explanation is attenuated, if not entirely misplaced in the context in which it is published, especially in Maclean’s, and has political impact that you didn’t imagine would occur.

Your mystification over “misreading” is perhaps excusable naïveté. What would perhaps be a greater fault would be if you teach this material without balancing it with the various alternate viewpoints.

[…] about the next part: can we maintain our standard of living, or even enhance it, indefinitely while extracting finite res… As Queen’s economist John Hartwick taught us years ago, the answer depends on three things: […]